When there is only one input signal
, the input vector
in Fig.2.28 can be defined as the scalar input
times a
vector of gains:

where
is an matrix. Similarly, a single output can
be created by taking an arbitrary linear combination of the
components of
. An example single-input, single-output (SISO)
FDN for
is shown in Fig.2.29.

Note that when
, this system is capable of realizing
anytransfer function of the form

By elementary state-space analysis [452, Appendix E],
the transfer function can be written in terms of the FDN system parameters
as

where
denotes the
identity matrix. This is easy
to show by taking the z transform of the impulse response of the system.

The more general case shown in Fig.2.29 can be handled in one of
two ways: (1) the matrices
can be augmented
to order
such that the three delay lines are replaced
by
unit-sample delays, or (2) ordinary state-space analysis
may be generalized to non-unit delays, yielding

In FDN reverberation applications,
, where
is an orthogonal matrix, for reasons addressed below, and
is a
diagonal matrix of lowpass filters, each having gain bounded by 1. In
certain applications, the subset of orthogonal matrices known as
circulant matrices have advantages [388].